On the number of critical points of stable solutions in bounded strip-like domains

In this paper we show that there exists a family of domains Omega(epsilon) subset of R-N with N >= 2, such that the stable solution of the problem{-Delta u = g(u) in Omega(epsilon) u>0 in Omega(epsilon) u=0 in Omega(epsilon)admits k critical points with k >= 2. Moreover the sets Omega(epsilon) are star-shaped and "close" to a strip as epsilon -> 0. Next, if g(u) 1and N >= 3we exhibit a family of domains Omega(epsilon) with positive mean curvature and solutions u(epsilon) which have k critical points with k >= 2. In this case, the domains Omega(epsilon) turn out to be "close" to a cylinder as e -> 0. (C) 2021 Elsevier Inc. All rights reserved.